How to fix inconsistent behavior of two quads defined by the same corner vertices?
Edit (Better understanding of OP)
If you show the internal triangulation for rendering by using a Wireframe node, you can see that the (non-planar) quads are indeed triangulated in opposite directions:

..(and here the shipped add-on Measureit has been used to show vertex indices at a reasonable size.)
I've found that can be flipped by CtrlT manually triangulating with a real, editable edge, selecting the appropriate direction, 'Fixed' for 1st and 3rd vertices, 'Fixed Alternate' for 2nd and 4th:

Then CtrlX Limited Dissolving the new edge, setting the angle high enough to lose it:

After that, the internal triangulation appears to follow the hint:

.. and you get good ol' Z-fighting between the coincident faces, which is what you would expect.